EFFECTIVE PROPERTIES OF POWER-LAW SOLIDS CONTAINING ELLIPTIC INHOMOGENEITIES .1. RIGID INCLUSIONS

The plane strain deformation of a power-law material containing a dispersion of rigid elliptical inclusions is investigated. Accurate constitutive relations are established for dilute concentrations of inclusions over a wide range of martrix hardening exponents and inclusion aspect ratios. The essential step in the analysis is the solution of the kernel problem for an isolated inclusion, and this is obtained using a Ritz procedure in which trial displacements are derived from a displacement potential in elliptic-cylindrical coordinates. Approximate constitutive relations for nondilute concentrations of inclusions are then established using a differential self-consistent scheme. The study is primarily concerned with randomly oriented inclusions, but limited results are also presented for aligned inclusions. As part of the investigation, the procedure introduced by Ponte Castaneda (J. Mech. Phys. Solids (1991)) is evaluated. This is a variational procedure which exploits information about a linear solid to obtain bounds or estimates for the overall response of nonlinear solids with the same microgeometry of second phase. Finally, connection is made with recent results for spheroidal inclusions given by Lee and Mear (J. Mech. Phys. Solids (1991); Int. J. Solids Struct. (1991)), and an ad hoc estimate for the overall response of solids containing randomly oriented rigid, spheroidal inclusions is proposed.